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On a Family of Functions Defined Over Sums of Primes
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Christian Axler

Department of Mathematics

Heinrich-Heine-University

40225 Düsseldorf

Germany

**Abstract:**

Let *r* and *m* be real numbers so that the sum
*S*_{r,m}(*x*) =
∑_{p ≤ x} *p*^{r}
log^{m} *p* diverges
as *x* → ∞. Here *p* runs
over all primes not exceeding *x*.
In this paper, we give an
asymptotic formula for each
*S*_{r,m}(*x*)
as *x* → ∞. The case where *x*
is the *n*th prime
number is of particular interest.
Here we use a method developed by Salvy to give an
asymptotic formula for
*S*_{r,m}(*p*_{n})
as *n* → ∞,
which generalizes, for instance, the previously
known one for
*S*_{1,0}(*p*_{n}),
the sum of the first *n* prime numbers.

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(Concerned with sequences
A000040
A007504.)

Received December 17 2018; revised versions received December 19 2018; March 29 2019;
July 8 2019.
Published in *Journal of Integer Sequences*,
August 23 2019.

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